https://file.aiquickdraw.com/mathbot/file/17266530544745xwgeb3n.jpg

The expression provided in the image is:

\[
\frac{4x^2 - 11x + 6}{16x^2 - 9}
\]

### Step 1: Factor the numerator and the denominator

#### Numerator: \( 4x^2 - 11x + 6 \)

We need to factor this quadratic expression. We look for two numbers that multiply to \( 4 \times 6 = 24 \) and add to \( -11 \). The numbers are \( -3 \) and \( -8 \).

So we rewrite \( -11x \) as \( -3x - 8x \):

\[
4x^2 - 11x + 6 = 4x^2 - 3x - 8x + 6
\]

Now factor by grouping:

\[
= (4x^2 - 8x) - (3x - 6)
\]
\[
= 4x(x - 2) - 3(x - 2)
\]
\[
= (4x - 3)(x - 2)
\]

#### Denominator: \( 16x^2 - 9 \)

This is a difference of squares:

\[
16x^2 - 9 = (4x - 3)(4x + 3)
\]

### Step 2: Simplify the expression

Now we have:

\[
\frac{(4x - 3)(x - 2)}{(4x - 3)(4x + 3)}
\]

Cancel the common factor \( (4x - 3) \) from the numerator and the denominator:

\[
\frac{x - 2}{4x + 3}
\]

### Final Answer:

The simplified form of the expression is:

\[
\frac{x - 2}{4x + 3}
\]

Would you like further details or explanations? Here are some related questions:

1. How do you factor a quadratic expression like \( 4x^2 - 11x + 6 \)?
2. What is the difference of squares and how is it factored?
3. Why can you cancel out terms like \( 4x - 3 \) in a rational expression?
4. How would the expression change if it were \( 16x^2 + 9 \) instead of \( 16x^2 - 9 \)?
5. What if the quadratic in the denominator couldn't be factored? How would you approach it?

**Tip:** Always check for common factors when simplifying algebraic expressions, but make sure that any term you're canceling is a factor, not just part of a term!

Simplify the expression (4x^2 - 11x + 6) / (16x^2 - 9).

Learn how to simplify the rational expression (4x^2 - 11x + 6) / (16x^2 - 9) by factoring the numerator and the denominator. This step-by-step solution covers key algebraic concepts, including quadratic equations, factoring techniques, and the difference of squares theorem. Suitable for students in Grades 9-11.

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